Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

Abstract

The modelling of diffusive terms in particle methods is a delicate matter and several models were proposed in the literature to take such terms into account. The diffusion velocity method (DVM), originally designed for the diffusion of passive scalars, turns diffusive terms into convective ones by expressing them as a divergence involving a so-called diffusion velocity. In this paper, DVM is extended to the diffusion of vectorial quantities in the three-dimensional Navier–Stokes equations, in their incompressible, velocity–vorticity formulation. The integration of a large eddy simulation (LES) turbulence model is investigated and a DVM general formulation is proposed. Either with or without LES, a novel expression of the diffusion velocity is derived, which makes it easier to approximate and which highlights the analogy with the original formulation for scalar transport. From this statement, DVM is then analysed in one dimension, both analytically and numerically on test cases to point out its good behaviour.

Appendices

Appendix A: Vectorial DVM developments

The aim of this appendix is to show that $\mathrm {B}$ can be expressed as in Eq. (16). First, from the definition (8), it follows that:

$$\begin{aligned} B_{ij} = \nu \frac{\partial \omega _j}{\partial x_i} - \nu \frac{\partial \omega _p}{\partial x_i} \frac{\omega _p\omega _j}{|\varvec{\omega }|^2} \end{aligned}$$

(55)

were repeated indices (here $p$) indicate a sum over those indices. Using the unit vector $\mathbf {e}=\varvec{\omega }/|\varvec{\omega }|$, the previous equation becomes

$$\begin{aligned} B_{ij} = \nu \frac{\partial \omega _j}{\partial x_i} - \nu \frac{\partial \omega _p}{\partial x_i} e_pe_j \end{aligned}$$

(56)

One can then make the following transformations:

$$\begin{aligned} \frac{\partial \omega _p}{\partial x_i} e_pe_j&= \frac{\partial \omega _pe_pe_j}{\partial x_i} - \omega _p \frac{\partial e_pe_j}{\partial x_i} = \frac{\partial \omega _je_pe_p}{\partial x_i} - \omega _pe_p \frac{\partial e_j}{\partial x_i} - \omega _pe_j \frac{\partial e_p}{\partial x_i} \end{aligned}$$

(57)

$$\begin{aligned}&= \frac{\partial \omega _j}{\partial x_i} - |\varvec{\omega }| \frac{\partial e_j}{\partial x_i} - \omega _je_p \frac{\partial e_p}{\partial x_i}, \end{aligned}$$

(58)

whose last term vanishes since $e_p \partial e_p/\partial x_i = \frac{1}{2} \partial (e_pe_p)/\partial x_i = 0$. Finally, $B_{ij}$ becomes:

$$\begin{aligned} B_{ij} = \nu |\varvec{\omega }| \frac{\partial e_j}{\partial x_i}. \end{aligned}$$

(59)

Appendix B: Analytic expressions of the Gaussian kernel and its Fourier transform

The purpose of this appendix is to determine the explicit expressions of the one-dimensional Gaussian kernel and its Fourier transform, for any order of accuracy. The approach is mostly inspired by reproducing kernel particle method (RKPM) techniques [see for example Liu et al. (1995, 1996)]. One should remember that the following derivation is restricted to Gaussian kernels, and aims at providing expressions that are explicit and simpler than the more general recurrence relations proposed in the existing literature.

1.1 Notations

Let $\zeta _{m, \varepsilon }$ be the interpolation kernel of order $r=2(m+1)$, where $\varepsilon$ is the mollifying parameter, issuing from the function $\zeta _m$ [cf. Eq. (28)]. Since the kernel $\zeta _{m, \varepsilon }$ (resp. its Fourier transform ${\hat{\zeta }}_{m, \varepsilon }$) can easily be deduced from $\zeta _m$ (resp. ${\hat{\zeta }}_{m}$), whose expression is much simpler, the proof will be performed on $\zeta _m$ and ${\hat{\zeta }}_{m}$. The 2^nd order function $\zeta _{0}$ is defined by:

$$\begin{aligned} \zeta _{0}(x) = \frac{1}{\sqrt{\pi }} \mathrm{e}^{-x^2}. \end{aligned}$$

(60)

It follows that

$$\begin{aligned} {\hat{\zeta }}_{0}(k) = \mathrm{e}^{-k^2/4}. \end{aligned}$$

(61)

1.2 Hermite polynomials

Let $H_n$ be the $n$th Hermite polynomial, whose degree is $n$, $n\ge 0$. $H_n$ is classically defined by:

$$\begin{aligned} \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}} \mathrm{e}^{-x^2} = (-1)^n \mathrm{e}^{-x^2} H_n(x), \quad \forall n \ge 0. \end{aligned}$$

(62)

These polynomials have an explicit expression, which is here split up according to the parity of $n$:

$$\begin{aligned} H_{2n}(x)&= (2n)! \sum \limits _{p=0}^n \frac{(-1)^{n-p}}{(2p)! (n-p)!} (2x)^{2p}, \quad \forall n \ge 0. \end{aligned}$$

(63a)

$$\begin{aligned} H_{2n+1}(x)&= (2n+1)! \sum \limits _{p=0}^n \frac{(-1)^{n-p}}{(2p+1)! (n-p)!} (2x)^{2p+1}, \quad \forall n \ge 0. \end{aligned}$$

(63b)

1.3 Hermite polynomials with parameter $a$

Hermite polynomials can easily be generalized to Gaussian functions with scaling parameter $a>0$. Let $H^*_n(x,a)$ be the $n$-th generalized Hermite function of parameter $a$, defined by:

$$\begin{aligned} \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}} \mathrm{e}^{-ax^2} = (-1)^n \mathrm{e}^{-ax^2} H^*_n(x,a), \quad \forall n \ge 0, \, \forall a>0. \end{aligned}$$

(64)

It is related to $H_n$ by the following relation:

$$\begin{aligned} H^*_n(x,a) = a^{n/2} H_n(\sqrt{a} x), \quad \forall n \ge 0, \, \forall a>0. \end{aligned}$$

(65)

1.4 Hermitian linear combination

The $r$th order function $\zeta _m$ is built from a Gaussian template as follows (Eldredge et al. 2002):

$$\begin{aligned} \zeta _m(x)= \frac{\mathrm{e}^{-x^2}}{\sqrt{\pi }} \sum _{n=0}^{m} \gamma ^{[m]}_n x^{2n}. \end{aligned}$$

(66)

where the $\gamma ^{[m]}_n$ are $m$ related coefficients determined by the moment conditions. This means that $\zeta _m$ is the product of $\frac{\mathrm{e}^{-x^2}}{\sqrt{\pi }}$ with an even polynomial (i.e. of even powers only) of degree $2m$. The proof relies on the fact that $\zeta _m$ can be written as a linear combination of the $m+1$ first even derivatives of $\zeta _{0}$ (including the zero-th derivative):

$$\begin{aligned} \zeta _m(x) = \sum _{n=0}^{m} c_n \zeta _{0}^{(2n)}(x). \end{aligned}$$

(67)

As a matter of fact, $\zeta _{0}^{(2n)}$ are even polynomials of degree $2n$ multiplied by the Gaussian function $\frac{\mathrm{e}^{-x^2}}{\sqrt{\pi }}$. It can be easily shown that any family of $m+1$ even polynomials of degrees $0$ to $2m$ forms a basis for the $2m$th degree even polynomials.

By taking the Fourier transform of (67), one obtains:

$$\begin{aligned} {\hat{\zeta }}_{m}(k) = \sum _{n=0}^{m} c_n \mathcal {F} \left( \frac{\mathrm{d}^{2n}}{\mathrm{d}x^{2n}} \zeta _{0}(x) \right) = {\hat{\zeta }}_{0}(k) \sum _{n=0}^{m} (-1)^n c_n k^{2n}. \end{aligned}$$

(68)

1.5 Moment conditions

The moment conditions [cf., amongst others, Beale and Majda (1985), Liu et al. (1995), Eldredge et al. (2002)] are:

$$\begin{aligned}&\displaystyle \int \zeta _m(y) \, \mathrm{d}y = 1, \end{aligned}$$

(69a)

$$\begin{aligned}&\displaystyle \int y^{p} \zeta _m(y) \, \mathrm{d}y = 0, \quad \forall 1 \le p \le r-1 . \end{aligned}$$

(69b)

However, thanks to the choice of even functions (cf. Eq. (66)), odd moment conditions are naturally respected. Moment conditions (69b) thus reduce to:

$$\begin{aligned} \int y^{2p} \zeta _m(y) \, \mathrm{d}y = 0, \quad \forall 1 \le p \le m . \end{aligned}$$

(70)

In the Fourier domain, those conditions are, respectively, equivalent to:

$$\begin{aligned}&\displaystyle {\hat{\zeta }}_{m}(0) = 1, \end{aligned}$$

(71a)

$$\begin{aligned}&\displaystyle {\hat{\zeta }}_{m}^{(2p)}(0) = 0, \quad \forall 1 \le p \le m . \end{aligned}$$

(71b)

1.6 Expression of the $c_n$

Using (68) evaluated at $k=0$, it follows directly from (71a) that $c_0=1$.

In addition, for $1 \le p \le m$, ${\hat{\zeta }}_{m}^{(2p)}(k)$ is given by:

$$\begin{aligned} {\hat{\zeta }}_{m}^{(2p)}(k) = \sum _{n=0}^{m} \left( -1\right) ^n c_n {\hat{\phi }}_{p,n}(k), \quad \text{ with } {\hat{\phi }}_{p,n}(k)=\frac{\mathrm{d}^{2p}}{\mathrm{d}k^{2p}} \left( k^{2n} {\hat{\zeta }}_{0}(k) \right) \!. \end{aligned}$$

(72)

• If $n>p$, ${\hat{\phi }}_{p,n}(0) = 0$ and thus

  $$\begin{aligned} {\hat{\zeta }}_{m}^{(2p)}(0) = \sum _{n=0}^{p} (-1)^n c_n {\hat{\phi }}_{p,n}(0). \end{aligned}$$

  (73)

• If $n \le p$, using Leibniz rule,

  $$\begin{aligned} {\hat{\phi }}_{p,n}(k) = \sum _{j=0}^{2p} \left( {\begin{array}{c}2p\\ j\end{array}}\right) \frac{\mathrm{d}^{j}}{\mathrm{d}k^{j}}\left( k^{2n} \right) \frac{\mathrm{d}^{2p-j}}{\mathrm{d}k^{2p-j}} \left( {\hat{\zeta }}_{0}(k) \right) \!, \end{aligned}$$

  (74)

  which, evaluated at $k=0$, reduces to

  $$\begin{aligned} {\hat{\phi }}_{p,n}(0) = \left( {\begin{array}{c}2p\\ 2n\end{array}}\right) (2n)! \, {\hat{\zeta }}_{0}^{\left( 2(p-n) \right) }(0) = \frac{(2p)!}{\left[ 2(p-n) \right] !} {\hat{\zeta }}_{0}^{\left( 2(p-n) \right) }(0). \end{aligned}$$

  (75)

  Thanks to Hermite polynomials, it follows that, for any $q>0$,

  $$\begin{aligned} {\hat{\zeta }}_{0}^{2q}(0) = \left( \frac{\mathrm{d}^{2q}}{\mathrm{d}k^{2q}} \mathrm{e}^{-k^2/4} \right) _{k=0} = \left( \frac{1}{2^{2q}} \mathrm{e}^{-k^2/4} H_{2n}(k) \right) _{k=0} = \frac{(-1)^q}{2^{2q}} \frac{(2q)!}{q!} \end{aligned}$$

  (76)

  and, combined with Eqs. (75) and (76), Eq. (73) becomes

  $$\begin{aligned} {\hat{\zeta }}_{m}^{(2p)}(0) = \frac{(-1)^p (2p)!}{2^{2p}} \sum _{n=0}^{p} \frac{2^{2n}}{(p-n)!} c_n . \end{aligned}$$

  (77)

Condition (71b) implies

$$\begin{aligned} \sum _{n=0}^{p} \frac{2^{2n}}{(p-n)!} c_n = 0, \quad \forall 1\le p \le m, \end{aligned}$$

(78)

which eventually leads to the following recursive relation:

$$\begin{aligned} c_p = -\frac{1}{2^{2p}} \sum _{n=0}^{p-1} \frac{2^{2n}}{(p-n)!} c_n, \quad \forall 1 \le p \le m . \end{aligned}$$

(79)

By a simple recursive induction, it can be shown that the sequence $(c_n)_{n \ge 0}$ has the following explicit definition:

$$\begin{aligned} c_n = \frac{(-1)^n}{2^{2n} n!}, \quad \forall 0 \le n \le m . \end{aligned}$$

(80)

1.7 Consequences on the kernel and its Fourier transform

Injecting (80) into (67) leads to

$$\begin{aligned} \zeta _m(x) = \sum _{n=0}^m \frac{(-1)^n}{2^{2n} n!} \zeta _{0}^{(2n)}(x), \end{aligned}$$

(81)

and, using Hermite polynomials to explicitly express $\zeta _{0}^{(2n)}(x)$, a general analytic expression for a Gaussian kernel of any order $r$ is obtained:

$$\begin{aligned} \zeta _m(x) = \frac{\mathrm{e}^{-x^2}}{\sqrt{\pi }} \sum _{p=0}^m \left[ (-1)^p \sum _{n=p}^m \beta _p^{[n]} \right] x^{2p}, \end{aligned}$$

(82)

where

$$\begin{aligned} \beta _p^{[n]} = \frac{(2n)!}{(2p)!} \frac{1}{2^{2(n-p)} \, n! (n-p)!} \in \mathbb {Q}, \quad \forall p=0,\ldots ,m. \end{aligned}$$

(83)

This explicit expression is not very helpful as such. However, it follows from (81) that $\zeta _m$ is defined from $\zeta _{m-1}$ and the $2m$-th derivative of $\zeta _{0}$:

$$\begin{aligned} \zeta _m(x)=\zeta _{m-1}(x)+ \frac{(-1)^m}{2^{2m}m!} \zeta _{0}^{(2m)}(x). \end{aligned}$$

(84)

Using again Hermite polynomials to explicitly express $\zeta _{0}^{(2m)}(x)$, the values of the $\gamma ^{[m]}_n \in \mathbb {Q}$, which entirely define $\zeta _m$, can thus be incrementally induced from the previous order kernel:

$$\begin{aligned} \left\{ \begin{aligned} \gamma ^{[m]}_n&= \gamma ^{[m-1]}_n + (-1)^n \beta _n^{[m]} \quad \forall n=0,\ldots ,m-1 \\ \gamma ^{[m]}_m&= (-1)^m \beta _m^{[m]} = \frac{(-1)^m}{m!}. \end{aligned} \right. \end{aligned}$$

(85)

In fact, the explicit expression (82) is more interesting and useful in the Fourier domain. By injecting (80) into (68), one obtains:

$$\begin{aligned} {\hat{\zeta }}_{m}(k) = \mathrm{e}^{-k^2/4} \sum _{n=0}^m \frac{k^{2n}}{2^{2n} n!}. \end{aligned}$$

(86)

and thus

$$\begin{aligned} {\hat{\zeta }}_{m, \varepsilon }(k) = \mathrm{e}^{-\varepsilon ^2 k^2/4} \sum _{n=0}^m \frac{1}{n!} \left( \frac{\varepsilon }{2}\right) ^{2n} k^{2n}. \end{aligned}$$

(87)

Those results are very interesting in the sense that they provide explicit expressions of Gaussian kernels of any order of accuracy. In addition, their Fourier transform have an incremental explicit definition and satisfy the condition ${\hat{\zeta }}_{m}(k) \le {\hat{\zeta }}_{m}(0)$, which is the condition required by Cortez (1997) for stability.

1.8 Particle approximation

In the sequel, the particle approximation of the Gaussian patch $c(x,t)= \exp \left( -x^2/(4\nu \,t)\right) /\sqrt{4\pi \nu \, t}$, presented in Sect. 4, is analysed.

In an effort to simplify notations, the time dependence of some functions may be omitted in the sequel, although it should be present both in the function and the Fourier domains. Let $\sigma =\sqrt{4\nu t}$ and $\alpha =\sqrt{\varepsilon ^2 + \sigma ^2}$. Then

$$\begin{aligned} {\hat{\zeta }}_{m, \varepsilon } \hat{c}(k) = \mathrm{e}^{-\alpha ^2 k^2/4} \sum _{n=0}^m \frac{1}{n!} \left( \frac{\varepsilon }{2}\right) ^{2n} k^{2n}, \end{aligned}$$

(88)

and thus

$$\begin{aligned} \langle c \rangle (x) = \mathcal {F}^{-1} \left[ {\hat{\zeta }}_{m, \varepsilon } \hat{c} \right]&= \sum _{n=0}^m \frac{1}{n!} \left( \frac{\varepsilon }{2}\right) ^{2n} \mathcal {F}^{-1} \left[ k^{2n} \mathrm{e}^{-\alpha ^2 k^2/4} \right] \end{aligned}$$

(89)

$$\begin{aligned}&= \frac{1}{\alpha \sqrt{\pi }} \sum _{n=0}^m \frac{(-1)^n}{n!} \left( \frac{\varepsilon }{2}\right) ^{2n} \frac{d^{2n}}{dx^{2n}} \mathrm{e}^{-x^2/\alpha ^2} . \end{aligned}$$

(90)

Using Hermite polynomials,

$$\begin{aligned} \langle c \rangle (x) = \frac{1}{\alpha \sqrt{\pi }} \mathrm{e}^{-x^2/\alpha ^2} P_m(x), \end{aligned}$$

(91)

with

$$\begin{aligned} P_m(x) = \sum _{p=0}^m \frac{(-1)^p}{(2p)!} \left( \sum _{n=p}^m \frac{(2n)! \, \varepsilon ^{2n}}{2^{2(n-p)} n! \, (n-p)! \, \alpha ^{2(n+p)}} \right) x^{2p} . \end{aligned}$$

(92)

Likewise, since $\langle \nabla c \rangle =\mathcal {F}^{-1} \left[ ik {\hat{\zeta }}_{m, \varepsilon } \hat{c} \right]$ it follows that:

$$\begin{aligned} \langle \nabla c \rangle (x) = \frac{-2x}{\alpha ^3 \sqrt{\pi }} \mathrm{e}^{-x^2/\alpha ^2} Q_m(x), \end{aligned}$$

(93)

with

$$\begin{aligned} Q_m(x) = \sum _{p=0}^m \frac{(-1)^p}{(2p+1)!} \left( \sum _{n=p}^m \frac{(2n+1)! \, \varepsilon ^{2n}}{2^{2(n-p)} n! \, (n-p)! \, \alpha ^{2(n+p)}} \right) x^{2p} . \end{aligned}$$

(94)

1.9 Consequences on the error

The analytic exact expressions of $c$, $\nabla c$ and $u_d$ are given by:

$$\begin{aligned} c(x)&=\frac{1}{\sigma \sqrt{\pi }}\mathrm{e}^{-x^2/\sigma ^2}, \end{aligned}$$

(95)

$$\begin{aligned} \nabla c(x)&=\frac{-2x}{\sigma ^3\sqrt{\pi }}\mathrm{e}^{-x^2/\sigma ^2}, \end{aligned}$$

(96)

and

$$\begin{aligned} u_d(x)=\frac{x}{2t}. \end{aligned}$$

(97)

It follows that

$$\begin{aligned} \frac{\langle c \rangle }{c}(x)&= \frac{\sigma }{\alpha } \mathrm{e}^{\varepsilon ^2 x^2/\alpha ^2} P_m(x), \end{aligned}$$

(98)

$$\begin{aligned} \frac{\langle \nabla c \rangle }{\nabla c}(x)&= \frac{\sigma ^3}{\alpha ^3} \mathrm{e}^{\varepsilon ^2 x^2/\alpha ^2} Q_m(x), \end{aligned}$$

(99)

and

$$\begin{aligned} \frac{\langle u_d \rangle }{u_d}(x) = \frac{\sigma ^2}{\alpha ^2} \frac{Q_m}{P_m}(x) . \end{aligned}$$

(100)

Expressing $\mathcal {E}_{\langle c \rangle }$, $\mathcal {E}_{\langle \nabla c \rangle }$ and $\mathcal {E}_{\langle u_d \rangle }$ from these last three expressions is straightforward.

It should be noted that $P_m$ and $Q_m$ being polynomials with degree $2m$, the error $\mathcal {E}_{\langle u_d \rangle }$ is spatially constant for the 2nd order ($r=2$ and thus $m=0$) approximation, at any given time $t>0$.